Send the Gift of Lifelong Learning!

Great Thinkers, Great Theorems

Gifting Information

Frequently Asked Questions

With an eGift, you can instantly send a Great Course to a friend or loved one via email. It's simple:

Find the course you would like to eGift.

Under “Choose a Format,” click on Video Download or Audio Download.

Click ‘Send e-Gift’

Fill out the details on the next page. You will need the email address of your friend or family member.

Proceed with the checkout process as usual.

Once you have paid for your order, your friend or loved one will receive an email letting them know that they have a gift waiting for them at TheGreatCourses.com. That gift will be added to their My Digital Library when they log in and click to redeem it.

Why do I need to specify the email of the recipient?

We will send that person an email to notify them of your gift.

If they are already a customer, they will be able to add the gift to their My Digital Library and mobile apps.

If they are not yet a customer, we will help them set up a new account so they can enjoy their course in their My Digital Library or via our free mobile apps.

How will my friend or family member know they have a gift?

They will receive an email from The Great Courses notifying them of your eGift. The email will direct them to TheGreatCourses.com.

What if my friend or family member does not receive the email?

If the email notification is missing, first check their Spam folder. Depending on your email provider, it may have mistakenly been flagged as spam. If it is not found, please email customer service at (customerservice@thegreatcourses.com) or call 1-800-832-2412 for assistance.

How will I know they have received my eGift?

When the recipient clicks on their email and redeems their eGift, you will automatically receive an email notification.

I don’t want to send downloads. How do I gift DVDs or CDs?

eGifting only covers digital products. To purchase a DVD or CD version of a course and mail it to a friend, please call customer service at 1-800-832-2412 for assistance.

Oops! The recipient already owns the course I gifted. What now?

Great minds think alike! We can exchange the eGifted course for another course of equal value. Please call customer service at 1-800-832-2412 for assistance.

Can I select a date in the future to send my eGift?

Sorry, this feature is not available yet. We are working on adding it in the future.

What if the email associated with eGift is not for my regular Great Course account?

Please email customer service at (customerservice@thegreatcourses.com) or call our customer service team at 1-800-832-2412 for assistance. They have the ability to update the email address.

When purchasing a gift for someone, why do I have to create an account?

This is done for two reasons. One is so you can track the purchase of the order in your ‘order history’ section as well as being able to let our customer service team track your purchase and the person who received it if the need arises.

Can I return or Exchange a gift after I purchase it?

Because the gift is sent immediately, it cannot be returned or exchanged by the person giving the gift. The recipient can exchange the gift for another course of equal or lesser value, or pay the difference on a more expensive item

Frequently Asked Questions

With an eGift, you can instantly send a Great Course to a friend or loved one via email. It's simple:
1. Find the course you would like to eGift.
2. Under "Choose a Format", click on Video Download or Audio Download.
3. Click 'Send e-Gift'
4. Fill out the details on the next page. You will need to the email address of your friend or family member.
5. Proceed with the checkout process as usual.

Q: Why do I need to specify the email of the recipient?

A:
We will send that person an email to notify them of your gift. If they are already a customer, they will be able to add the gift to their My Digital Library and mobile apps. If they are not yet a customer, we will help them set up a new account so they can enjoy their course in their My Digital Library or via our free mobile apps.

Q: How will my friend or family member know they have a gift?

A:
They will receive an email from The Great Courses notifying them of your eGift. The email will direct them to TheGreatCourses.com. If they are already a customer, they will be able to add the gift to their My Digital Library and mobile apps. If they are not yet a customer, we will help them set up a new account so they can enjoy their course in their My Digital Library or via our free mobile apps.

Q: What if my friend or family member does not receive the email?

A:
If the email notification is missing, first check your Spam folder. Depending on your email provider, it may have mistakenly been flagged as spam. If it is not found, please email customer service at (customerservice@thegreatcourses.com) or call 1-800-832-2412 for assistance.

Q: How will I know they have received my eGift?

A:
When the recipient clicks on their email and redeems their eGift, you will automatically receive an email notification.

Q: What if I do not receive the notification that the eGift has been redeemed?

A:
If the email notification is missing, first check your Spam folder. Depending on your email provider, it may have mistakenly been flagged as spam. If it is not found, please email customer service at (customerservice@thegreatcourses.com) or call customer service at 1-800-832-2412 for assistance.

Q: I don't want to send downloads. How do I gift DVDs or CDs?

A:
eGifting only covers digital products. To purchase a DVD or CD version of a course and mail it to a friend, please call customer service at 1-800-832-2412 for assistance.

Q: Oops! The recipient already owns the course I gifted. What now?

A:
Great minds think alike! We can exchange the eGifted course for another course of equal value. Please call customer service at 1-800-832-2412 for assistance.

Q: Can I update or change my email address?

A:
Yes, you can. Go to My Account to change your email address.

Q: Can I select a date in the future to send my eGift?

A:
Sorry, this feature is not available yet. We are working on adding it in the future.

Q: What if the email associated with eGift is not for my regular Great Course account?

A:
Please please email customer service at (customerservice@thegreatcourses.com) or call our customer service team at 1-800-832-2412 for assistance. They have the ability to update the email address so you can put in your correct account.

Q: When purchasing a gift for someone, why do I have to create an account?

A:
This is done for two reasons. One is so you can track the purchase of the order in your ‘order history’ section as well as being able to let our customer service team track your purchase and the person who received it if the need arises.

Q: Can I return or Exchange a gift after I purchase it?

A:
Because the gift is sent immediately, it cannot be returned or exchanged by the person giving the gift. The recipient can exchange the gift for another course of equal or lesser value, or pay the difference on a more expensive item

What are priority codes?

Priority Codes are on the back of the catalog, mail promotion, or within an advertisement. To ensure that the pricing on the website is the same as what is in your catalog or advertisement, please enter the priority code provided.

What are priority codes?

Priority Codes are on the back of the catalog, mail promotion, or within an advertisement. To ensure that the pricing on the website is the same as what is in your catalog or advertisement, please enter the priority code provided.

Course Overview

Mathematics is filled with beautiful theorems that are as breathtaking as the most celebrated works of art, literature, or music. They are the Mona Lisas, Hamlets, and Fifth Symphonys of the field—landmark achievements that repay endless study and that are the work of geniuses as fascinating as Leonardo, Shakespeare, and Beethoven. Here is a sample:

Pythagorean theorem: Although he didn't discover the Pythagorean theorem about a remarkable property of right triangles, the Greek mathematician Euclid devised an ingenious proof that is a mathematical masterpiece. Plus, it's beautiful to look at!

Area of a circle: The formula for the area of a circle, A = π r2, was deduced in a marvelous chain of reasoning by the Greek thinker Archimedes. His argument relied on the clever tactic of proof by contradiction not once, but twice.

Basel problem: The Swiss mathematician Leonhard Euler won his reputation in the early 1700s by evaluating an infinite series that had stumped the best mathematical minds for a generation. The solution was delightfully simple; the path to it, bewilderingly complex.

Larger infinities: In the late 1800s, the German mathematician Georg Cantor blazed the trail into the "transfinite" by proving that some infinite sets are bigger than others, thereby opening a strange new realm of mathematics.

You can savor these results and many more in Great Thinkers, Great Theorems, 24 half-hour lectures that conduct you through more than 3,000 years of beautiful mathematics, telling the story of the growth of the field through a carefully chosen selection of its most awe-inspiring theorems.

Approaching great theorems the way an art course approaches great works of art, the course opens your mind to new levels of math appreciation. And it requires no more than a grasp of high school mathematics, although it will delight mathematicians of all abilities.

Your guide on this lavishly illustrated tour, which features detailed graphics walking you through every step of every proof, is Professor William Dunham of Muhlenberg College, an award-winning teacher who has developed an artist's eye for conveying the essence of a mathematical idea. Through his enthusiasm for brilliant strategies, novel tactics, and other hallmarks of great theorems, you learn how mathematicians think and what they mean by "beauty" in their work. As added enrichment, the course guidebook has supplementary questions and problems that allow you to go deeper into the ideas behind the theorems.

An Innovative Approach to Mathematics

Professor Dunham has been taking this innovative approach to mathematics for over a quarter-century—in the classroom and in his popular books. With Great Thinkers, Great Theorems you get to watch him bring this subject to life in stimulating lectures that combine history, biography, and, above all, theorems, presented as a series of intellectual adventures that have built mathematics into the powerful tool of analysis and understanding that it is today.

In the arts, a great masterpiece can transform a genre; think of Claude Monet's 1872 canvas Impression, Sunrise, which gave the name to the Impressionist movement and revolutionized painting. The same is true in mathematics, with the difference that the revolution is permanent. Once a theorem has been established, it is true forever; it never goes out of style. Therefore the great theorems of the past are as fresh and impressive today as on the day they were first proved.

What Makes a Theorem Great?

A theorem is a mathematical proposition backed by a rigorous chain of reasoning, called a proof, that shows it is indisputably true. As for greatness, Professor Dunham believes the defining qualities of a great theorem are elegance and surprise, exemplified by these cases:

Elegance: Euclid has a beautifully simple way of showing that any finite collection of prime numbers can't be complete—that there is always at least one prime number left out, proving that the prime numbers are infinite. Dr. Dunham calls this one of the greatest proofs in all of mathematics.

Surprise: Another Greek, Heron, devised a formula for triangular area that is so odd that it looks like it must be wrong. "It's my favorite result from geometry just because it's so implausible," says Dr. Dunham, who shows how, 16 centuries later, Isaac Newton used algebra in an equally surprising route to the same result.

Great Thinkers, Great Theorems includes many lectures that are devoted to a single theorem. In these, Professor Dunham breaks the proof into manageable pieces so that you can follow it in detail. When you get to the Q.E.D.—the initials traditionally ending a proof, signaling quod erat demonstrandum (Latin for "that which was to be demonstrated")—you can step back and take in the masterpiece as a whole, just as you would with a painting in a museum.

In other lectures, you focus on the biographies of the mathematicians behind these masterpieces—geniuses who led eventful, eccentric, and sometimes tragic lives. For example:

Cardano: Perhaps the most bizarre mathematician who ever lived, the 16th-century Italian Gerolamo Cardano was a gambler, astrologer, papal physician, convicted heretic, and the first to publish the solution of cubic and quartic algebraic equations, which he did after a no-holds-barred competition with rival mathematicians.

Newton and Leibniz: The battle over who invented calculus, the most important mathematical discovery since ancient times, pitted Isaac Newton—mathematician, astronomer, alchemist—against Gottfried Wilhelm Leibniz— mathematician, philosopher, diplomat. Each believed the other was trying to steal the credit.

Euler: The most inspirational story in the history of mathematics belongs to Leonhard Euler, whose astonishing output barely slowed down after he went blind in 1771. Like Beethoven, who composed some of his greatest music after going deaf, Euler was able to practice his art entirely in his head.

Cantor: While Vincent van Gogh was painting pioneering works of modern art in France in the late 1800s, Georg Cantor was laying the foundations for modern mathematics next door in Germany. Unappreciated at first, the two rebels even looked alike, and both suffered debilitating bouts of depression.

Describing a common reaction to the theorems produced by these great thinkers, Professor Dunham says his students often want to know where the breakthrough ideas came from: How did the mathematicians do it? The question defies analysis, he says. "It's like asking: ‘Why did Shakespeare put the balcony scene in Romeo and Juliet? What made him think of it?' Well, he was Shakespeare. This is what genius looks like!" And by watching the lectures in Great Thinkers, Great Theorems, you will see what equivalent genius looks like in mathematics.

Hide Full Description

24 lectures

| Average 30 minutes each

1

Theorems as Masterpieces

Certain theorems stand out as great masterpieces of mathematics that can be appreciated as great works of art. After hearing Professor Dunham explain this approach, discover the two ways of proving a theorem: direct proof and indirect proof. Also, meet some of the great thinkers whose ideas you will be studying. x

2

Mathematics before Euclid

Investigate three non-Greek civilizations that had robust traditions in mathematics. Then encounter a pair of Greek mathematicians who predated Euclid, but who left very deep footprints: Thales and Pythagoras—the latter renowned for the theorem that bears his name. x

3

The Greatest Mathematics Book of All

Begin your exploration of the work widely considered the greatest mathematical text of all time: Euclid's Elements. Discover why these 13 succinct books have been so influential for so long as you delve into the ground-laying definitions, postulates, common notions, and theorems from book I. x

4

Euclid's Elements—Triangles and Polygons

Continuing your journey through Euclid, work your way toward his most famous result: his proof of the Pythagorean theorem—a demonstration of remarkable visual and intellectual beauty. Also, cover some of the techniques from book IV for constructing regular polygons. x

5

Number Theory in Euclid

In addition to being a geometer, Euclid was a pioneering number theorist, a subject he took up in books VII, VIII, and IX of the Elements. Focus on his proof that there are infinitely many prime numbers, which Professor Dunham considers one of the greatest proofs in all of mathematics. x

6

The Life and Works of Archimedes

Even more distinguished than Euclid was Archimedes, whose brilliant ideas took centuries to fully absorb. Probe the life and famous death of this absent-minded thinker, who once ran unclothed through the streets, shouting "Eureka!" ("I have found it!") on solving a problem in his bath. x

7

Archimedes' Determination of Circular Area

See Archimedes in action by following his solution to the problem of determining circular area—a question that seems trivial today but only because he solved it so simply and decisively. His unusual strategy relied on a pair of indirect proofs. x

8

Heron's Formula for Triangular Area

Heron of Alexandria (also called Hero) is known as the inventor of a proto-steam engine many centuries before the Industrial Revolution. Discover that he was also a great mathematician who devised a curious method for determining the area of a triangle from the lengths of its three sides. x

9

Al-Khwarizmi and Islamic Mathematics

With the decline of classical civilization in the West, the focus of mathematical activity shifted to the Islamic world. Investigate the proofs of the mathematician whose name gives us our term "algorithm": al-Khwarizmi. His great book on equation solving also led to the term "algebra." x

10

A Horatio Algebra Story

Visit the ruthless world of 16th-century Italian universities, where mathematicians kept their discoveries to themselves so they could win public competitions against their rivals. Meet one of the most colorful of these figures: Gerolamo Cardano, who solved several key problems. In secret, of course. x

11

To the Cubic and Beyond

Trace Cardano's path to his greatest triumph: the solution to the cubic equation, widely considered impossible at the time. His protégé, Ludovico Ferrari, then solved the quartic equation. Norwegian mathematician Niels Abel later showed that no general solutions are possible for fifth- or higher-degree equations. x

12

The Heroic Century

The 17th century saw the pace of mathematical innovations accelerate, not least in the introduction of more streamlined notation. Survey the revolutionary thinkers of this period, including John Napier, Henry Briggs, René Descartes, Blaise Pascal, and Pierre de Fermat, whose famous "last theorem" would not be proved until 1995. x

13

The Legacy of Newton

Explore the eventful life of Isaac Newton, one of the greatest geniuses of all time. Obsessive in his search for answers to questions from optics to alchemy to theology, he made his biggest mark in mathematics and science, inventing calculus and discovering the law of universal gravitation. x

14

Newton's Infinite Series

Start with the binomial expansion, then turn to Newton's innovation of using fractional and negative exponents to calculate roots—an example of his creative use of infinite series. Also see how infinite series allowed Newton to approximate sine values with extraordinary accuracy. x

15

Newton's Proof of Heron's Formula

Return to Heron's ancient formula for determining the area of a triangle to consider Newton's proof using algebraic techniques—an approach he also applied to other geometry problems. The steps are circuitous, but the result bears Newton's stamp of genius. x

16

The Legacy of Leibniz

Probe the career of Newton's great rival, Gottfried Wilhelm Leibniz, who came relatively late to mathematics, plunging in during a diplomatic assignment to Paris. In short order, he discovered the "Leibniz series" to represent π, and within a few years he invented calculus independently of Newton. x

17

The Bernoullis and the Calculus Wars

Follow the bitter dispute between Newton and Leibniz over priority in the development of calculus. Also encounter the Swiss brothers Jakob and Johann Bernoulli, enthusiastic supporters of Leibniz. Their fierce sibling rivalry extended to their competition to outdo each other in mathematical discoveries. x

18

Euler, the Master

Meet history's most prolific mathematician, Leonhard Euler, who went blind in his sixties but kept turning out brilliant papers. A sampling of his achievements: the number e, crucial in calculus; Euler's identity, responsible for the most beautiful theorem ever; Euler's polyhedral formula; and Euler's path. x

19

Euler's Extraordinary Sum

Euler won his spurs as a great mathematician by finding the value of a converging infinite series that had stumped the Bernoulli brothers and everyone else who tried it. Pursue Euler's analysis through the twists and turns that led to a brilliantly simple answer. x

20

Euler and the Partitioning of Numbers

Investigate Euler's contribution to number theory by first warming up with the concept of amicable numbers—a truly rare breed of integers until Euler vastly increased the supply. Then move on to Euler's daring proof of a partitioning property of whole numbers. x

21

Gauss—the Prince of Mathematicians

Dubbed the Prince of Mathematicians by the end of his career, Carl Friedrich Gauss was already making major contributions by his teen years. Survey his many achievements in mathematics and other fields, focusing on his proof that a regular 17-sided polygon can be constructed with compass and straightedge alone. x

22

The 19th Century—Rigor and Liberation

Delve into some of the important trends of 19th-century mathematics: a quest for rigor in securing the foundations of calculus; the liberation from the physical sciences, embodied by non-Euclidean geometry; and the first significant steps toward opening the field to women. x

23

Cantor and the Infinite

Another turning point of 19th-century mathematics was an increasing level of abstraction, notably in the approach to the infinite taken by Georg Cantor. Explore the paradoxes of the "completed" infinite, and how Cantor resolved this mystery with transfinite numbers, exemplified by the transfinite cardinal aleph-naught. x

24

Beyond the Infinite

See how it's possible to build an infinite set that's bigger than the set of all whole numbers, which is itself infinite. Conclude the course with Cantor's theorem that the transcendental numbers greatly outnumber the seemingly more abundant algebraic numbers—a final example of the elegance, economy, and surprise of a mathematical masterpiece. x

Lecture Titles

Clone Content from Your Professor tab

What's Included

What Does Each Format Include?

Instant Video Includes:

Download 24 video lectures to your computer or mobile app

Downloadable PDF of the course guidebook

FREE video streaming of the course from our website and mobile apps

DVD Includes:

24 lectures on 4 DVDs

96-page printed course guidebook

Downloadable PDF of the course guidebook

FREE video streaming of the course from our website and mobile apps

What Does The Course Guidebook Include?

Course Guidebook Details:

96-page printed course guidebook

Photos & illustrations

Charts & diagrams

Timeline

Enjoy This Course On-the-Go with Our Mobile Apps!*

iPhone + iPad

Android Devices

Kindle Fire Tablet + Firephone

*Courses can be streamed from anywhere you
have an internet connection. Standard carrier data rates may apply in areas that do not have wifi connections
pursuant to your carrier contract.

Your professor

About Your Professor

William Dunham, Ph.D.

Muhlenberg College

Dr. William Dunham is the Truman Koehler Professor of Mathematics at Muhlenberg College in Allentown, Pennsylvania. He earned his undergraduate degree from the University of Pittsburgh and his M.S. and Ph.D. in Mathematics from The Ohio State University. Before his current appointment at Muhlenberg, Dr. Dunham taught at Hanover College in Indiana, receiving teaching awards from both institutions as well as the Award for...

Reviews

Great Thinkers, Great Theorems is rated
4.8 out of
5 by
87.

Rated 5 out of
5 by
johnthe geologist from
Fun for everyoneI have been familiar with the work of Prof. Dunham for years after reading his books Journey through Genius and The Mathematical Universe. When I discovered he had a Great Courses series, I had to watch it. I was not disappointed.
He is a very entertaining man as well as an excellent speaker. His presentations were easy enough for me to understand even after 40 years since my last college calculus class. I hope he will do more.

Date published: 2019-08-15

Rated 5 out of
5 by
GeorgeGru from
Great course!Even though I majored in Applied Mathematics, I learned quite a bit from this course. The instructor was amazing in his explanations and sense of humor.

Date published: 2019-03-21

Rated 5 out of
5 by
robcatjrt from
EssentialIf you enjoy the history of Mathematics, this collection of lectures is a must-have for your library.

Date published: 2019-03-07

Rated 5 out of
5 by
Gord from
Amazing accomplishmentsTook me a few lectures to get into the rhythm of the subject again (age 58... been a few years since my last math class) but then I was hooked.
Professor does a fine job of both presenting the thinker AND the thinking.
I found it very engaging, understandable and well-paced.
The video would seem a “must”, in order to fully benefit from the careful step-by-step proofs presented.
Math proofs really can be fun, and sharpens your own thinking, too.

Date published: 2019-02-13

Rated 5 out of
5 by
rabäckdal from
What A Trip!If you like to read the 'Foreward' and 'Introduction' parts of a book to get the setting of what you are about to experience, then you will really enjoy taking this ride through the timeline of mathematical history, progress, and other exciting math discoveries. The professor stops at appropriate points and lets one get a view of who's doing what and why it is important. We get acquainted with personalities behind the names and even some of their little quirks. We become aware of how geographically vast was the interest in mathematical thinking. And the professor shares some of the unexpected delight in elegant results to profound problems - 'the mathematical stuff'. To top it all, Professor Dunham knows how to relate well with his students. As to intended audience, it helps to have something more than basic knowledge of mathematics but the professor sometimes goes to great depth to make sure he doesn't lose anyone regardless of background. This is a great course. I wish there had been more stops, its that good.

Date published: 2019-01-21

Rated 5 out of
5 by
garyc7 from
good coverage of great men & great discovereys3 December 2018
Great Courses Great Thinkers, Great Theorems #1471
Great coverage of major math people & ideas. Well presented, limited graphics, but OK
Review the Tale of contents to see the scope of coverage.
The guide book is limited in scope & value. It should have more detailed info.
Presenter is OK, nothing exceptional, but OK.

Date published: 2018-12-04

Rated 5 out of
5 by
RLPENDLETON from
A WONDERFUL COURSEI have purchased many of your courses. I am an engineer and know math at a very high level. This course introduced me to many things I had not studied. It gave me an appreciation of Euclid and euclidian thinking I had not had. The selection and presentation of these great ideas and great thinkers was truly extraordinary. I recommend this most highly

Date published: 2018-11-29

Rated 5 out of
5 by
KurtJSchmucker from
Wonderful topics with great presentationI enjoyed this Great Course immensely. Excellent topic and great presentations.